Local solutions and existence of optimal transport maps for the $W_\infty$ Wasserstein distance. Extensions to more classical Monge problems.
MSRI Summer Microprogram on Nonlinear Partial Differential Equations
July 24, 2007 04:00 PM to 04:50 PM
Speakers:
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Abstract: |
I will consider the non-linear optimal transportation problem of minimizing the cost functional $\C_\infty(\lambda)= \lambda\text{-}\esssup_{(x,y) \in \Omega^2} |y-x|$ in the set of probability measures on $\Omega^2$ having prescribed marginals.
This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. I will establish the existence of ``local'' solutions and characterize this class with aid of an adequate version of cyclical monotonicity.
Moreover, under natural assumptions, I will show that local solutions are induced by transport maps.
The lack of duality forces us to introduce a different technique to prove the existence of an optimal transport map. At the end of the talk I will show how this technique may be usefull to tackle the classical Monge problem for some norms. (Joint paper with T.Champion and P.Juutinen, work in progress with T. Champion.) |
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Lecture #12437
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